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Just so that we can on the same page, I will present the couple definitions, let $X$ be the underlying metric space

'Bolzano-Weierstrass Property' is when every bounded sequence in $X$ has a converging subsequence.

Sequentially compact is when every sequence in $X$ has a converging subsequence.

Of course, sequentially compact is stronger than 'Bolzano-Weierstrass Property' but are there occasions where BWP will imply sequentially compact?

Bernard
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UnsinkableSam
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BWP implies sequential compactness iff $X$ is itself is bounded. This is because an unbounded sequence has a subsequence which has no convergent subsequence.

Suppose $(x_n)$is unbounded. Fix a point $x$. Then there is a subsequence $(x_{k_n})$ such that $d(x_{k_n},x) >n$ for all $n$. If this subsequence has subsequence converging to some $y$ we get $d(x,y)=\infty$, a contradiction.

Kavi Rama Murthy
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  • I am not sure that an unbounded sequence can never have a convergent subsequence. Take $x_n=0$ for even $n$ and $x_n=n$ for odd $n$. Clearly $x_n$ is unbounded above, but the subsequence $x_{2n}$ is constant and therefore convergent. – Math1000 Nov 03 '19 at 00:10
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    @Math1000 Thank you. You are right. I have edited my answer. – Kavi Rama Murthy Nov 03 '19 at 00:21